Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme

hibitively costly in terms of computational time. Instead, I suggest that the joint ch.f. be inverted using
the FFT algorithm, and a kernel-smoothing scheme be used to mitigate the problem of precision loss. A
several orders of magnitude reduction in the time of the inversion, however, comes at a cost of dramatically
increasing demand for computer memory. A fast FT algorithm that would produce satisfactory results will
not be “fast” on an average modern PC. It is more efficient to obtain p.d.f.’s separately on a server with large
RAM and use the results to price options on a, possibly, less powerful machine.

To assess the numerical accuracy of the method, I apply it to price all SAP U.S. 100 index options ( “OEX” )
that expire in 2004 and are quoted at market’s closing on 7 consecutive trading days, starting with June 30^th.
Parameters are calibrated on “XEO” options and July 9^th data is set aside for out-of-the-sample predictions.
Pricing errors overall look reasonable and out-of-the-sample errors are not fundamentally different from the
in-sample ones. Still, the method tends to underprice out-of-the-money puts and predictions are relatively
coarse for options with “long” (∣ of the year) time to expiration. The latter result is, most likely, an artifact
of the degrading accuracy of the linear Richardson extrapolation.

References

[1] Barone-Adesi, Giovanni, Robert E. Whaley. 1987. “Efficient Analytic Approximation of American Option
Values,” Journal of Finance, Vol. 42, No. 2 (June), 301-320

[2] Bates, David S. 1996. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche
Mark Options,” Review of Financial Studies, Vol. 9, No.1 (Spring), 69-108

[3] Billingsley, Patrick. 1995. Probability and Measure. 3rd ed. John Wiley A Sons Ltd.

[4] Broadie, Mark, Jerome Detemple. 1996. “American Option Valuation: New Bounds, Approximations,
and a Comparison of Existing Methods,” Review of Financial Studies, Vol. 9, No. 4 (Winter), 1211-1250

[5] Chernov, Mikhail, Eric Ghysels. 2000. “A Study Towards a Unified Approach to the Joint Estimation
of Objective and Risk Neutral Measures for the Purpose of Options Valuation,” Journal of Financial
Economics. Vol. 56, 407-458

[6] Chicago Board of Options Exchange. 2004a. Europeanstyle S&P 100 Index Options: Product Specifica-
tions. http://www.cboe.com/OptProd/indexopts/xeo_spec.asp

[7] Chicago Board of Options Exchange. 2004b. OEX S&P 100 Index Options: Product Specifications.
http://www.cboe.com/OptProd/indexopts/oex_spec.asp

[8] Cox, John C., Jonathan E. Ingersoll, Stephen A. Ross. 1985. “A Theory of the Term Structure of Interest
Rates,” Econometrica, Vol. 53, No. 2 (March), 385-408

[9] Duffie Darrell, Jun Pan, and Kenneth Singleton. 2000. “Transform Analysis and Asset Pricing for Affine
Jump-Diffusions,” Econometrica. Vol. 68, No. 6 (November), 1343-1376

[10] Epps, Thomas W. 2004a. Econ 83.fi Derivative Securities. UVA

[11] Epps, Thomas W. 2004b. Option Pricing Under Stochastic Volatility with Jumps. UVA, mimeo

[12] Feller, William. 1951. “Two Singular Diffusion Problems,” Annals of Mathematics, Vol. 54, No. 1 (July),
173-182

18

<<   17   18   19   20   21   22   23   >>

More intriguing information